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Effects of soil stiffness and depth on the seismic response of sites in total and effective stress analyses Reza Imam & Danial Ghaffarian Department of Civil and Environmental Engineering Amirkabir University of Technology, Tehran, Iran ABSTRACT Many building codes require site response analysis for the design of buildings on sites underlain by very weak or liquefiable soils. In such cases, response spectra curves and seismic design parameters are determined based on the response of the site soils to earthquake loading. Site response is most commonly studied using one-dimensional, effective or total stress analyses. Various factors that influence results of the analyses should be considered in order to arrive at design parameters that envelope all possible cases. In the current study, site response analyses are carried out for a site underlain by liquefiable soils. A response spectra curve corresponding to maximum considered earthquake (MCE) is obtained from ground motion seismic hazard analysis. This curve, developed for site conditions compatible with the base of the soil column used in the site response analysis, is used for the selection of seven acceleration time history records consistent with the spectra shape, the site properties, and its seismicity. The records are then spectrally matched to this target curve, and then applied at the base of the soil column. Effects of variability in soil stratigraphy and stiffness, and depth of soil column on results of the analyses are examined. Due to liquefaction potential of the subsoil, effects of pore pressure generation and dissipation are examined by conducting nonlinear effective stress analyses, and results are compared to those obtained from total stress analyses. Results indicate that stiffer stratigraphy lead to higher spectral accelerations at lower periods, and softer stratigraphy result in higher accelerations in higher periods. Smaller depth of soil column result in higher spectral accelerations at ground surface. Moreover, due to inability of liquefiable soils to transfer shear stresses to the upper layers, smaller spectral accelerations are obtained at ground surface from effective stress analyses. 1 INTRODUCTION Seismic design of buildings is mainly based on the intensity of ground shaking due to earthquake loads at ground surface. This intensity depends, among other factors, on the stratigraphy, stiffness, and depth of the subsoil that transfer the earthquake shaking from the stiff soil or rock at depth to the usually softer soils at ground surface.

In estimating the intensity of shaking at ground surface, analysis of site response to earthquake loads using one-dimensional (1D) analysis of upward propagation of seismic waves, assuming free-field condition is commonly used. In order to arrive at results that can be used in safe design of buildings, such analyses should take into account the various site characteristics that affect the response of the site to ground shaking, including subsoil stratigraphy, stiffness of soil layers, depth to competent soil, and generation and dissipation of pore pressures, where applicable. Most of these naturally occurring site characteristics are associated with some variability over the area of the site considered, and the site response analyses should take into account these variations in order to ensure that the most critical condition is considered in design. Moreover, where site soils are susceptible to loss of stiffness and strength due to liquefaction, such changes in soil properties should also be taken into account in the site response analysis.

1.1 Some previous studies on site response Numerous studies on the effect of various factors on response of project sites to earthquake shaking have been carried out in the past, a summary of some of the more recent studies is provided below.

Macra et al. (2005) compared results obtained from 1D and 2D models used for the analysis of Mygdonian basin and studied the effect of site shape. Badaoui et al. (2007) used boundary element method for studying effects of depth of layer thickness in a multi-layer soil and the effect of presence of a tunnel on the seismic response of the site. Rota et al. (2010) used 1D equivalent linear analysis to study effects of soil layer parameters on the response of the Ancona site in Itay subjected to various earthquake loads. Davoodi et al. (2013) used non-linear finite difference method to study the response of an earth dam to near-field and far-field earthquake loading. Hashash et al. (2015) studied the accuracy of estimating soil surface response obtained from 1D analysis using centrifuge physical model. Markham et al. (2016) compared results of 1D and 2D finite difference analyses in order to evaluate the efficiency of 1D models in the analysis of site responses considering pore pressure generation during earthquake loading.

. 1.2 The current study

Limited studies conducted in the past examined the sensitivity of site response to soil stratigraphy, properties and depth, using total and effective stress analysis taking into account pore pressure generation and dissipation of liquefiable soils. The current study employs the more commonly used 1D analysis to obtain and compare results of non-linear site response analysis for upper-bound, lower bound and average soil stiffnesses as determined from shear wave velocity measurements, and for two different depths of competent soil layer. Due to the liquefaction potential of the subsoil, effective stress analysis with pore pressure generation and dissipation is also carried out for the case of average soil stiffness and results are compared with those from total stress analysis. Finally, response spectra curves and amplification values that can be used for design of the project, taking into account envelope of results obtained from the various analysis cases, are obtained 2 SITE SUBSOIL CONDITIONS 2.1 Subsoil stratigraphy The subsoils consist predominantly of silts and clayey silts with some layers of sands in the order of five (5) feet (1.5 m) thick to depths of forty-six (46) to fifty five (55) feet (14 to 16.75 m). Based on the SPT and CPT results, the upper forty-six (46) to fifty-five (55) feet (14 to 16.75 m) below existing grade consist of loose to medium dense, coarse-grained soils or soft to stiff, fine-grained soils. Below this depth, up to the maximum depth of investigation of approximately seventy-five (75) feet (22.85 m), the subsoil consists primarily of dense to very dense sands with occasional thin interbeds of silty sands and silt. At depths greater than approximately seventy- five (75) feet (22.85 m), penetration refusal was generally encountered due to the presence of very dense and gravelly sand. The clays are of "low" plasticity (CL), with plasticity index of typically less than twelve (12). 2.2 Measured shear wave velocities Shear wave velocities were measured to a maximum depth of seventy-five (75) feet (22.85 m) at two locations (SCPT-1 and SCPT-7) and soil stratigraphy based on CPT and SPT test results were also obtained up to this maximum depth at other locations within the site. Based on these results, average, upper limit and lower limit soil stratigraphy profiles were selected as shown in Figure 1. These profiles were used in the site response analyses in order to consider sensitivity of the results to soil stratigraphy. Sensitivity of results to depth of soil model was also examined by comparing results from soil models with fifty five (55) and seventy-five (75) feet (16.75 and 22.85 m) depth. 2.3 Liquefaction potential of the subsoil Liquefaction analysis of the site soils indicated the presence of liquefiable layers to a depth of fifty five (55) feet (16.75 m) below ground surface. Considering that the

estimated natural period of the proposed building is approximately 1.2 seconds, the site is considered class F according to the ASCE7-10 standard, and site response analysis is required for the determination of seismic design parameters. For the liquefaction analyses carried out, a historic high groundwater depth of eight (8) feet (2.45 m) below existing ground surface was used based on records of past measurements of ground water depths near the project site.

Figure 1. Shear wave velocity profiles for the cases analyzed 3 SITE RESPONSE ANALYSIS 3.1 Method of analysis Site response analysis was carried out using the computer program Deepsoil ver. 6.1 (Hashash et.al., 2012). This program can be used to conduct 1D non-linear site response analyses assuming vertically propagating shear waves that are transmitted to a soil column due to loading at its base. The input base motions propagates vertically through the idealized soil column to the ground surface and it is assumed that the soil layers are horizontal. In such analyses, effects of vertical motions of the base soil or rock, surface waves, lateral changes in thickness and depth of soil layers, and changes in the properties of the soil layer

in the horizontal direction are not considered (Kavazanjian et al. 1997).

Since depth of groundwater at the project site was determined to be more than seventy (70) feet (21.33 m), most analyses were carried out using total stresses, and did not consider effects of groundwater. However, due to the liquefaction potential of the subsoil, and the historic high groundwater depth of eight (8) feet (2.45 m) used in the liquefaction analysis, effective stress analysis with pore water pressure generation and dissipation was also conducted in addition to the common total stress analyses.

Parameters and information needed for site response analysis include stratigraphy, unit weight and stiffness of each of the soil layers, changes in stiffness and damping ratio of the soils with strain, and depth of bedrock or competent soil. 3.2 Input parameters Based on results of laboratory tests, for the total and effective stress analyses carried out, the variations of unit weight of the subsoil as shown in Figure 2 were used for the various stratigraphies used in the analses.

Figure 2. Variation of soil unit weight with depth: (a) For average and lower bound shear wave velocities – water table at 75 ft. (22.85 m) depth, (b) For upper bound shear wave velocities – water table at 75 ft. (22.85 m) depth and (c) For average shear wave velocities – water table at 8 ft. (2.45 m) depth.

As indicated before, three cases for the variation of shear wave velocity with depth, namely an upper bound, a lower bound, and an average profile were used in the site response analyses.

Maximum shear moduli at small strain (Gmax) may be calculated based on the shear wave velocities (Vs) and soil densities (ρ) for each profile using the following relationship: Gmax = ρ Vs2 [1]

No experimental data regarding the degradation of shear modulus and damping ratio with shear strain was available for the on-site soils. However, based on previous studies on local soils the Seed and Idriss (1990) and Vucetic and Dobry (1991) curves for variations of modulus reduction and damping ratio with shear strain were used for the sandy and silty soils, respectively, as shown in Figure 3. As commonly assumed in such analyses, the shear modulus and damping ratio for the base soil half space with shear wave velocity of 1,200 feet per second (360 m/s) was considered constant.

Figure 3. Shear moduli and Damping ratios 3.3 Analysis cases Six different cases of analyses were considered as described below and summarized in Table 1. Case (1) Total stress analysis of average soil profile with seventy-five (75) feet (22.85 m) depth. Case (2) Total stress analysis of upper bound profile with seventy-five (75) feet (22.85 m) depth. Case (3) Total stress analysis of lower bound soil profile with seventy-five (75) (22.85 m) feet depth. Case (4) Total stress analysis of average soil profile with fifty-five (55) feet (16.75 m) depth. Case (5) Effective stress analysis of average soil profile with fifty-five (55) feet (16.75 m) depth. Case (6) Effective stress analysis of average soil profile with seventy-five (75) feet (22.85 m) depth.

Table 1. Different cases of site response analysis

No. Description

1 Total stress - Lower limit soil stratigraphy - 75 ft. depth

2 Total stress - Average soil stratigraphy - 75 ft. depth

3 Total stress - Upper limit soil stratigraphy - 75 ft. depth

4 Total stress - Average soil stratigraphy - 55 ft. depth

5 Effective stress - Average soil stratigraphy - 75 ft. depth

6 Effective stress - Average soil stratigraphy - 55 ft. depth

4 INPUT MOTION 4.1 Base soil response spectra In order to select the acceleration time histories for use in the site response analysis, a response spectra curve corresponding to risk-targeted, Maximum Considered Earthquake (MCER) was determined for the project site. This curve was obtained following a site-specific seismic hazard analysis using the deterministic and probabilistic methods according to the ASCE 7-10 procedure. This was completed by considering the subsoil condition and seismicity of the site at the base soil level, where competent soil with minimum shear wave velocity of 1200 feet per second (360 m/s) exists.

Deaggregation of magnitudes and distances of the seismic sources for various spectral periods indicated that magnitudes of 6.5 to 7.1 and distances of 6.4 to 23 kilometers were applicable for the site. 4.2 Selection of acceleration time histories Using results of the seismic source deaggregation described before, and considering shape of the site-specific MCER response spectra, the shear wave velocity of the base soil, the earthquake source mechanism, and other site and seismicity parameters, seven recorded acceleration time histories consistent with the site seismicity and subsoil conditions were selected as shown in Table 2. Table 2. Time histories selected for site response analysis

Record No.

Event Name Event Year

Station Name Mag. Faulting

Mechanism

NGA#864 Landers 1992 Joshua Tree 7.28 Strike-Slip

NGA#1042 Northbridge-

01 1994

N Hollywood – Coldwater Can

6.69 Reverse

NGA#1082 Northbridge-

01 1994

Sun Valley – Roscoe Blvd

6.69 Reverse

NGA#4847 Chuestu-oki_Japan

2007 Joetsu

Kakizakiku 6.8 Reverse

NGA#5780 Iwate_Japa

n 2008 Iwadeyama 6.9 Reverse

NGA#6893 Darfield_New Zealand

2010 DFHS 7 Strike-Slip

NGA#6960 Darfield_New Zealand

2010 Riccarton High

Scool 7 Strike-Slip

4.3 Spectral matching of the selected time histories

Bazzurro and Luco (2006) indicated that spectral matching of records reduces the number of time histories needed for site response analysis. A number of methods may be used for modifying recorded acceleration time histories in order for their response spectra to match a specified target response spectra curve (Preumont, 1984). Each method may use one of the following approaches:

1- Spectral matching in the frequency domain 2- Spectral matching in the frequency domain and

using the theory of additional vibrations 3- Spectral matching in the time domain

Use of the third approach usually provides better convergence and better preserves the characteristics of the input motion. This is done by adding wavelets to the original time history such that its response spectra matches the target spectra.

Lilhanad and Tseng (1987, 1988) introduced the algorithm for spectral matching over the time domain. However, their algorithm can result in drift in the velocity and displacement time histories of the matched record. Al Atik et al. (2010) introduced an algorithm, which they indicated that it preserves the characteristics of the original time history in addition to preventing drift.

The seven selected time history records listed in Table 2 were spectrally matched to the target MCER spectra curve described before using the RspMatch99 program implemented in the EZ-Frisk 7.65 software, and then applied to the base of the soil column during the site response analyses. 5 ANALYSIS RESULTS 5.1 Total and effective stress analyses The Deepsoil computer program calculates time-histories of acceleration, velocity, and displacements resulting from the input ground motion applied at the base soil level. Sample results from the analyses is shown in Figure 4 for record No. 1082. Figure 4(a) compares input and output acceleration time histories for case 4 (in Table 1) and Figure 4(b) compares time histories for case 6. As may be noticed in these figures, for total stress analysis in which no pore pressure generation and dissipation is considered, variations of accelerations at ground surface nearly follows those of the input motion applied at base soil. However, for case 6 in which soil depth and stiffness profile is similar to those of case 4 but effective stress analysis is carried out, after some time passes from the start of seismic loading, the output motion at ground surface no longer follows the input motion at base soil. This is because at this time, the soil in the middle of the soil column liquefies, resulting in a reduction in its shear strength and inability to transfer the seismic loads to the ground surface. In this case, amplitude and frequency of the motion at ground surface decreases substantially compared to the input motion and to the results obtained from total stress analysis at ground surface.

Figure 4. Comparison of input and output acceleration time histories from site response analysis (a) Analysis case 4 and (b) Analysis case 6 Figure 5 shows maximum shear strains developed in the soil at various depths due to application of Record #1082. Maximum shear strains are highest in layers closer to the bedrock where they reach values close to 3%. However, they are lower at shallower depths and lowest near ground surface.

It is noted that result for each soil layer is provided by the software for the point at the top of the layer. Therefore, for cases 4 and 6 in which maximum depth of the soil column is 55 ft (16.75 m) and depth of the lowest layer varies from 35 to 55 ft (10.67 to 16.75 m), the lowest data point at which data is available is at 35 ft depth (10.67 m). For these two cases, therefore, plots are extended only up to this depth. For other cases in which maximum depth is 75 ft (22.85 m), plots are extended up to 65 ft (19.8 m), which is the depth of top of the lowest layer.

Figure 5. Maximum shear strain in the soil for record #1082 during different analysis cases

5.2 Factors affecting effective stress analysis

Relative similarity of input and output motions as shown in Figure 4(a) was observed in all other results of total stress analyses. However, output results from effective stress analyses depended on various factors. These factors include: duration of the input motion, total time of strong shaking at base soil, frequency and amplitude of the applied loads, and depth of the bedrock or competent soil. Properties of the soil at ground surface will also affect output accelerations transferred to the ground surface.

Figure 6 shows variations of excess pore water pressure ratio (Δu/σ’v) with depth and variations with time at seven depths, obtained from the effective stress analyses (cases 5 and 6) due to loading from record #1082. It may be noticed that a pore pressure ratio of unity (full liquefaction) occurs about 15 seconds after start of loading at depths below approximately 35 ft (10.67 m). These results are also confirmed by Figures 4 and 5 and indicate that after occurrence of liquefaction, only a small portion of the dynamic loads applied at base soil is transferred to the upper layers and, maximum shear strain and pore pressure ratios are much smaller at these depths compared to the lower layers at which soil is liquefied. Moreover, excess pore pressures and shear strains are higher for case 6, in which depth of competent soil is smaller, compared to case 5, as expected.

Figure 6. Excess pore water pressure generation from effective stress analyses

Figure 7 shows output time history results for case 6 obtained from applying the other six input motions, and Figure 8 shows results obtained from effective stress analyses in which record No 1082 is applied to soil profiles with different depths. Results from Figures 7 and 8 indicate that both input motion and depth of competent soil have significant influence on the motion at ground surface in liquefiable soils.

5.3 Response spectra at base soil and ground surface

Response spectra of the ground motions at ground surface obtained from applying the seven records and for the six cases described before are shown in Figure 9. These results indicate that as soil stiffness increases, the period at which peak values of the response acceleration occur decrease, and the peak value of the acceleration spectra increases. This may be noticed by comparing Figures 9 (a), (b) and (c). Moreover, decrease in depth of the competent soil results in increase in spectral accelerations in the low period range as shown in Figure 9(d).

Figure 7. Input and output accelerations, case 6: (a) NGA864–0 deg. (b)NGA1042–270 deg. (c)NGA4847–EW (d)NGA5780–NS (e)NGA6893–S17E (f)NGA6960–N86W.

Figure 8. Input and output accelerations from site response analysis (a) Analysis No.5 and (b) Analysis No.6.

Taking into account the pore pressure generation and dissipation during seismic loading in an effective stress analysis leads to decrease in spectral accelerations in the lower period range and its increase in the higher period range, as may be noticed from Figures 9 (e) and (f). These figures also indicate that results of effective stress analysis exhibit greater variations in response compared to results of total stress analysis, likely due to the various conditions that may arise with regard to the liquefaction of the soil layers.

5.4 Amplifications In order to gain an overall view of the results obtained from the various analysis cases, average response spectra obtained from applying the seven records is determined for each of the six analysis cases and results are shown in Figure 10. By dividing these averages to the response spectra at base soil for each period, amplifications of the base soil motions by the softer overburden soils are obtained for each analysis case and the results are plotted in Figures 11.

Results shown in Figure 11 indicate that in a total stress analysis, as depth of competent soil increases, or stiffness of the soil layers decreases, amplification decreases in the low period range and increases in the high period range. In the effective stress analysis, due to decrease in soil stiffness resulting from increase in pore pressure, soil behavior resembles that of softer soil profiles in which lower amplifications occur at low period ranges and higher amplifications occur at high period ranges. 6 SUMMARY AND CONCLUSIONS Results of analyses carried out during the current study indicate that 1D total and effective stress analyses of soils susceptible to liquefaction during cyclic loading lead to significantly different results. In order to obtain a realistic response of sites to seismic loading, it is not sufficient to carry out site response analysis using soil properties obtained from monotonic tests and take into account stiffness degradation at various shear strains. Instead, changes in pore pressures resulting from cyclic loading should also be taken into account and loss of stiffness and strength resulting from increase in pore pressure should also be considered in the analysis. Such loss of stiffness is expected to lead to lower spectral accelerations in the lower period range and higher accelerations in the higher period range. Moreover, due to inability of liquefiable soils to transfer shear stresses to the upper layers, smaller spectral accelerations are obtained at ground surface from effective stress analyses

It was also noticed that stiffer stratigraphy lead to higher spectral accelerations at lower periods, and softer stratigraphy result in higher accelerations in higher periods. Smaller depth of soil column results in higher spectral accelerations at ground surface.

It should be noted that results obtained from the current analyses were based on the use of the stiffness degradation and damping ratio curves selected in the current study. Use of other curves are expected to lead to

results that may be somewhat different from those obtained from this study, although the general conclusions are not expected to be affected.

Figure 9. MCER response spectra at base soil and ground surface (a) Analysis No.1, (b) Analysis No.2, (c) Analysis No.3, (d) Analysis No.4, (e) Analysis No.5 and (f) Analysis No.6.

Figure 10. Average response spectra at ground surface vs input MCER response spectra at soil base

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